Figure 6.30:

Phase portrait of conversion versus temperature at showing stable and unstable limit cycles.

Code for Figure 6.30

Text of the GNU GPL.

main.m

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
% Copyright (C) 2001, James B. Rawlings and John G. Ekerdt
%
% This program is free software; you can redistribute it and/or
% modify it under the terms of the GNU General Public License as
% published by the Free Software Foundation; either version 2, or (at
% your option) any later version.
%
% This program is distributed in the hope that it will be useful, but
% WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
% General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program; see the file COPYING.  If not, write to
% the Free Software Foundation, 59 Temple Place - Suite 330, Boston,
% MA 02111-1307, USA.
%
% Revised 8/13/2018

%
% limit cycle parameters
%
% units are kmol, min, kJ, K, m^3
%

p = struct();
p.k_m      = 0.004;
p.T_m      = 298;
p.E        = 15000;
p.c_Af     = 2;
p.C_p      = 4;
p.rho      = 1000;
p.C_ps     = p.C_p*p.rho;
p.T_f      = 298;
p.T_a      = p.T_f;
p.DeltaH_R = -2.2e5;
p.U        = 340;
%p.theta    = 72.3308;
p.theta    = 73.1;
p.T_set    = 321.53;
p.c_set    = 0.48995;
p.T_fs     = p.T_f;
p.Kc       = 0;
p.gamma    = p.E/p.T_f;
p.B        = -p.DeltaH_R*p.c_Af*p.gamma/(p.C_ps*p.T_f);
p.beta     = p.U/p.C_ps*p.theta;
p.Da       = p.k_m*exp(-p.E*(1/p.T_f-1/p.T_m))*p.theta;
p.x2c      = (p.T_a-p.T_f)/p.T_f*p.gamma;


% find the stable limit cycle
x0 = [(1-0.8)*p.c_Af 310];
tfinal = 40*p.theta;
ntimes = 3;
tout  = linspace(0, tfinal, ntimes);
opts = odeset('AbsTol', sqrt (eps), 'RelTol', sqrt (eps));
[tsolver, x] = ode15s(@(t, x) rhs(t, x, p), tout, x0, opts);

% now go around the limit cycle once with many time points
x0 = x(end,:);
ntimes = 200;
tfinal = 3*p.theta;
tout = linspace(0, tfinal, ntimes);
opts = odeset('AbsTol', sqrt (eps), 'RelTol', sqrt (eps));
[tsolver, x] = ode15s(@(t, x) rhs(t, x, p), tout, x0, opts);
u = (x(:,2) - p.T_set)*p.Kc + p.T_fs;
conv = (p.c_Af - x(:,1))/p.c_Af;
stablim = [tout' x conv u];

% reverse time and stabilize the unstable limit cycle starting in the
% interior


% find the unstable limit cycle
% 73.4957904330517 305.839679413557 0.516502467355002 -0.00139021373543844 0.0329582458327707 -0.00139021373543844 -0.0329582458327707 1
% stable steady state is about 0.5165, 305.83

x0=[(1-0.7)*p.c_Af; 305];
tfinal = 40*p.theta;
ntimes = 3;
tout   = linspace(0, tfinal, ntimes);
opts = odeset('AbsTol', sqrt (eps), 'RelTol', sqrt (eps));
[tsolver, x] = ode15s(@(t, x) reverserhs(t, x, p), tout, x0, opts);
% now go around the unstable limit cycle once with many time points
x0=x(end,:);
tfinal = 3*p.theta;
ntimes = 200;
tout   = linspace(0, tfinal, ntimes);
opts = odeset('AbsTol', sqrt (eps), 'RelTol', sqrt (eps));
[tsolver, x] = ode15s(@(t, x) rhs(t, x, p), tout, x0, opts);

u = (x(:,2) - p.T_set)*p.Kc + p.T_fs;
conv = (p.c_Af - x(:,1))/p.c_Af;
unstablim = [tout' x conv u];

table = [stablim unstablim];
save -ascii phase_portrait_2.dat table;
size(stablim)
size(unstablim)
disp(max(table(:,3)))
disp(max(table(:,4)))
disp(max(table(:,8)))
disp(max(table(:,9)))
if (~ strcmp (getenv ('OMIT_PLOTS'), 'true')) % PLOTTING
    plot (table(:,3),table(:,4),table(:,8),table(:,9));
    % TITLE
end % PLOTTING

rhs.m

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
function retval = rhs(t, x, p)
    c_A   = x(1);
    T     = x(2);
    k     = p.k_m*exp(-p.E*(1/T - 1/p.T_m));
    T_f   = p.T_fs + p.Kc*(T - p.T_set);

    retval = zeros (2, 1);
    retval(1) = (p.c_Af - c_A)/p.theta - k*c_A;
    retval(2) = p.U/p.C_ps*(p.T_a - T) + (p.T_f - T)/p.theta -...
                    k*c_A*p.DeltaH_R/p.C_ps;

reverserhs.m

1
2
3
4
5
6
7
8
9
function retval = reverserhs(t, x, p)
    c_A   = x(1);
    T     = x(2);
    k     = p.k_m*exp(-p.E*(1/T - 1/p.T_m));
    T_f   = p.T_fs + p.Kc*(T - p.T_set);
    retval = zeros (2, 1);
    retval(1) = -((p.c_Af - c_A)/p.theta - k*c_A);
    retval(2) = -(p.U/p.C_ps*(p.T_a - T) + (p.T_f - T)/p.theta -...
                    k*c_A*p.DeltaH_R/p.C_ps);